Variations

Variations of the Christoffel symbols, Riemann tensor, Ricci tensor, curvature scalar, and inverse metric:

(1)   \begin{eqnarray*} \delta \Gamma^\alpha_{\mu\nu} & = & \frac{1}{2}g^{\alpha\beta} \Bigl( \nabla_\mu \delta g_{\nu\beta} + \nabla_\nu \delta g_{\mu\beta} - \nabla_\beta \delta g_{\mu\nu} \Bigr) \\ \delta R^\alpha{}_{\beta\mu\nu} & = & \nabla_\mu \delta\Gamma^\alpha_{\beta\nu} - \nabla_\nu \delta\Gamma^\alpha_{\beta\mu} \\ \delta R_{\mu\nu} & = & \nabla_\alpha \delta\Gamma^\alpha_{\mu\nu} - \nabla_\mu \delta\Gamma^\alpha_{\nu\alpha} \\ \delta R & = & \nabla^\apha \nabla^\beta \delta g_{\alpha\beta} - \nabla^\alpha \nabla_\alpha \delta (\ln(-g)) - R^{\alpha\beta}\delta g_{\alpha\beta} \\ \delta g^{\alpha\beta} & = & -g^{\alpha\mu} \delta g_{\mu\nu}\,  g^{\nu\beta}  \end{eqnarray*}

where \nabla_\alpha is the covariant derivative compatible with the metric g_{\alpha\beta}. The variations \delta are general. They can be replaced by, say, coordinate derivatives or time derivatives. Bryce DeWitt’s favorite identity is

    \[ \delta(\ln(\det M)) = Tr(M^{-1} \delta M)  \]

where M is a matrix and Tr denotes the trace. Other useful relations, where S denotes a scalar, include

(2)   \begin{eqnarray*} \delta \nabla_\mu \nabla_\nu S & = & \nabla_\mu \nabla_\nu \delta S - \frac{1}{2} \nabla^\alpha S \Bigl( \nabla_\mu\delta g_{\nu\alpha} + \nabla_\nu\delta g_{\mu\alpha} - \nabla_\alpha\delta g_{\mu\nu} \Bigr) \\ \delta \nabla^2 S & = & \nabla^2 \delta S - (\nabla^\mu \nabla^\nu S) \delta g_{\mu\nu} - (\nabla^\mu S) \nabla^\nu\delta g_{\mu\nu} + (\nabla^\mu S) \nabla_\mu \delta (\ln\sqrt{-g})  \end{eqnarray*}